Transcript Document

EM Solvers Theory Basics
Basic theory review
The goal




Derive, from Maxwell field equations,
a circuital relationship bounding voltages
and currents
Give to the single terms of the circuit equation
the meaning of capacitance, inductance
and resistance
Explain the partial inductance concept
Summarize the high frequency effects
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Maxwell equations

Differential form of Maxwell equations:
 D  
B
 E  
t
B  0
D
 H  J 
t
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
The vector potential



By math, the divergence of the curl of a
vector is always zero:     A  0
Since   B  0
then we can write B    A
where Ais a magnetic vector potential
B
A
becomes   ( E 
 E  
)0
t
t
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
The scalar potential

A
  ( E  )  0 implies that there
t
exists a scalar function Φ such that
A
E
 
t
A
 We can rewrite it as
E 
 
t
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
The electric field (1)
A
 In E  
  the electric field
t
is expressed in terms of a scalar potential
and of a vector potential

Note that A is not uniquely defined; but
provided we are consistent with a gauge
choice, it does not matter here
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
The electric field (2)


Assume now that Ohm’s law is valid inside
the conductors: J  Etot
Let’s split Etot in E , induced by the charges
and currents in the system, and E0 , applied
by an external generator
Etot  E  E0
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
The electric field (3)
A
 Remembering E  
  and J  Etot ,
t
Etot  E  E0 can be rewritten as:
A
E0    

t
J

The applied field E0 is therefore split into
an ohmic term and terms due to the charges
and currents in the systems
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
The circuit equation

Consider a conductor mesh. For any point
A
along the path, E0    
is valid

t
J

To obtain a circuit equation, let’s integrate
this equation on the path:
A
E

dl


dl


dl




dl
0
c
c 
c t
c
J
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Kirkhoff’s second law

If
E
0
 dl
is defined as voltage applied
A
 dl     dl
to the circuit and   dl  

t
c
c
c
c
J
as voltage drops along the circuit, then
A
c E0  dl  c   dl  c t  dl  c   dl
J
has the form of the Kirkhoff’s second law
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Voltage drop terms

For low frequencies, i.e. small circuit
dimensions vs. wavelength (quasistatic
assumption), we can name the terms as:

c
applied
E0  dl 
voltage
A
’inductive’

dl

c t
voltage drop
internal
 dl  impedance

voltage drop
c

J
’capacitive’



dl


c
voltage drop
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Capacitance (1)



Reacting to the applied field, free charges
distribute along the circuit
Since all charges are regarded as field
sources, we can account for the conductor
through its effect on the field distribution
and assume that the charges are in free space
So the generated field is:

V
dV
4r
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Capacitance (2)


Assume to have a capacitor along the
chosen conductive path, as in the figure:
So along the path there is a discontinuity
(i.e. on the capacitor plates). Therefore:
2
   dl     dl     dl
c
1
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Capacitance (3)

Let’s work on the terms:
2
   dl     dl     dl
c
1
2
    dl  
   dl  0
1
(always zero)

Therefore,
1
   dl  
c
2
1

 dl  1  2
l
 2
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Capacitance (4)


By hypotesis, all charge Q is lumped on the
discontinuity
Φ is proportional to Q (remember  
so we can write
Q
1  2 
C
where C is constant
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved

V
dV
)
4r
Capacitance (5)

Considering the current continuity equation
Q  Idt , the charge on the discontinuity
is related with the current flowing towards it.

Q
1  2 
C



dl




1
2

Q
c  dl  C
c

Q
c  dl  C
is the usual capacitive term
of the circuit theory.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Inductance (1)


The A vector potential depends in general
on the current density J .
For example, assuming   A  0 , it follows:
A
V
JdV
4r
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Inductance (2)


The current density can be expressed as the
total current I multiplied by a suitable
coordinate-dependent vector function
along the conductor section.
Because of the quasistatic assumption,
I is constant along the circuit
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Inductance (3)

We can therefore define a coefficient, L,
depending on the circuit geometry but not
on the total current:
L
A

dl

c
I
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Inductance (4)

Using L, the ‘inductive’ voltage drop becomes:
A
d
d
dI
c t  dl  dt c A dl  dt ( LI )  L dt

This term has the usual form of inductive
voltage drop in the circuit theory
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Inductance (5)

As an alternative, consider a closed circuit,
and using Stoke’s theorem and the fact that
B    A we can define Ψ as:
A

dl

B

ds




S

So Ψ is the magnetic flux linked to the circuit;
this gives the usual inductance definition:
L
 A  dl
I

B

ds

S
I


I
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Internal impedance (1)



In general, the current is not homogeneously
distributed on a conductor section.
J
Therefore, the J term in    dl depends
c
strictly on the mesh integration path.
If we choose a path on the conductor
surface, J /  gives the electrical field
on the surface, ES .
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Internal impedance (2)

If we define the internal impedance Zi per
unit length as:
ES
Zi 
I
we can then write:
ES
c   dl  I C I  dl  I C Zi dl  IZ
J
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Internal impedance (3)

In sinusoidal steady state, Zi has a real and
an imaginary part, since the surface field is
not in phase with the total current in the
conductor, because of the magnetic flux
distribution inside the conductor.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Internal impedance (4)


The real part of Zi gives the resistance of the
conductor at a certain frequency.
The imaginary part of Zi gives the internal
reactance, that is the part of the reactance
generated by the magnetic flux inside
the conductor
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Internal impedance (5)


The choice of the integration path was
arbitrary; but in this way, the magnetic
flux Ψ in    A  dl  SB  ds can be
considered as the flux linked with the internal
of the mesh but not with the conductor.
Other choices could be made but with this
one there is a distinct advantage
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Internal impedance (6)

With this integration path choice:
L = Ψ / I can be called ‘external inductance’,
since is bounded to the magnetic flux
external to the conductor
The inductive term of the internal
impedance (the imaginary part) is linked
to the flux inside the conductor and
can be called ‘internal inductance’.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (1)



Inductance is a property of a closed
conductive path.
However, it is possible to define a partial
inductance, applicable to open paths
Suppose to divide a path into small,
rectilinear segments; the idea is to distribute
the total inductance to these segments
in a unique way
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (2)


Let’s work on a simple example. Consider the
rectangular path in figure:
Neglect the fact that the path is not closed;
assume also a constant current density
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (3)

Remembering the definition of magnetic flux,
   B  ds   A  dl we can define Ψ1, Ψ2,
S
Ψ3 and Ψ4 as integrals along the 4 segments:
4
4
    A  dl  i
i 1

seg i
i 1
This subdivision suggests that L  c A  dl / I
can written as the sum of four terms
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (4)

So, L=L1+ L2+ L3+ L4 where Li  segA  dl / I
i


The magnetic potential vector A can be
split in its turn into the sum of the potential
vectors generated by the currents in the
four sides
Assuming that the segments are
infinitesimally thin, we can now define the
partial inductances.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (5)

The partial inductances Lij are:
Lij


seg i
Aij  dli
Ij
where Aij is the vector potential along the
i segment caused by the current Ij along
the j segment
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (6)

The total inductance of the closed path is
therefore:
4
4
L   Lij
i 1 j 1

The definition can be extended to
non-infinitesimally thin segments with
non-uniform current densities
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (7)

Let’s consider an equivalent formula:
Lij


Si
Bij  dsi
Ij
where Si is the area enclosed between
segment i, the infinite and two lines
passing along the endpoints and
perpendicular to segment j
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (8)

For instance, if i = j, the area enclosed
in the path to be used in the integral is
shown in the figure:
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Partial inductance (9)

Let’s derive the alternative equation.
Using Stoke’s theorem on the path ci
enclosing Si, we have:
Lij


Si
Bij  dsi
Ij


ci
Aij  dli
Ij
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (10)

A has the following properties:
is parallel to the source current
increasing the distance from the source,
tends to zero
 By construction, lateral sides of Si are
perpendicular to source segment j

By construction, the last side of Si lies at
infinite distance
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
Partial inductance (11)

Therefore, the only contribution to Lij along
ci is given by the i segment, i.e.:
Lij



Si
Bij  dsi
Ij


ci
Aij  dli
Ij


seg i
Aij  dli
Ij
The equivalence has thus been proved.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
High frequency effects (1)


In this context, high freqency means the
maximum frequency at which we can still
use the quasistatic assumption
(i.e. geometrical dimensions << wavelength)
Three main effects:
Skin effect
Edge effect
Proximity effect
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
High frequency effects (2)



At low frequencies, the current is uniform
on the conductor section. Increasing the
frequency, the distribution changes and the
three effects take place.
The effects are not independent, but for
sake of simplicity are treated on their own
We will provide an intuitive explanation only
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
High frequency effects (3)



The edge effect is the tendency of the current
to crowd on the conductor edges.
The proximity effect, especially visible on
ground planes, is the tendency of the current
to crowd under signal carrying conductors.
The skin effect is the tendency of the current
to crowd on a thin layer (‘skin’) on the
conductor surface
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
High frequency effects (4)


The skin thickness is usually taken as δ,
the depth at which the field is only 1/e of
the surface field, in case of a plane conductor
The skin thickness formula is:

1
0 f
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
High frequency effects (5)
The proximity effect can be thought also
as the skin effect of a group of conductors
 At low frequencies, L and R are almost
constants. However, increasing the frequency,
the current crowds on the surface:

The resistance increases
The external inductance slightly decreases
The internal inductance decreases
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
High frequency effects (6)

At frequencies even higher, the R’ increases
with the square root of the frequency and
the L’ tends to a constant, which can be
expressed as:
L 
'
 0 0
C0
'
where C0’ is the capacitance per unit length
when the dielectrics are subsituted with void
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
References (1)
[1] S. Ramo, J. R. Whinnery, T. Van Duzer. Fields and Waves in Communication
Electronics. John Wiley & Sons Inc., 1993.
[2] A. E. Ruehli. Equivalent Circuit Models for Three-dimensional Multiconductor
Systems. IEEE Trans. on Microwave Theory and Techniques, vol. 22, no. 3,
March 1974.
[3] M. Kamon. Efficient Techniques for Inductance Extraction of Complex 3-D
Geometries. Master’s Thesis, Depart. of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, 1994.
[4] A. E. Ruehli. Inductance Calculations in a Complex Integrated Circuit
Environment. IBM J. Res. Develop., vol. 16, pp. 470-481, September 1972.
[5] E. Hallen. Electromagnetic Theory. Chapman & Hall, London, 1962.
[6] B. Young. Return Path Inductance in Measurements of Package Inductance
Matrixes. IEEE Trans. on Components, Packaging and Manufacturing
Technology, part B, vol. 20, no. 1, February 1997.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved
References (2)
[7] A. R. Djordjevic, T. K. Sarkar. Closed-Form Formulas for Frequency-Dependent
Resistance and Inductance per Unit Length of Microstrip and Strip
Transmission Lines. IEEE Trans. on Microwave Theory and Techniques,
vol. 42, no. 2, February 1994.
[8] S. S. Attwood. Electric and Magnetic Fields. John Wiley & Sons Inc., 1949.
Copyright 2012 Enrico Di Lorenzo, www.FastFieldSolvers.com, All Rights Reserved